A random point field on a locally compact Hausdorff space \(E\) with a measure \(\mu\) is considered. The appearance of \(n\) points of the field at \(n\) disjoint subsets of \(E\) is proposed to be determined by the \(n\)-dimensional density \(p_n\) with respect to \(\mu^n\). The author assumes that there exists an integral operator with a kernel \(K\) such that for any \(n\geq 2\) the density \(p_n\) is representable in the form of determinant of the kernel \(K\). Three theorems are proved. The first one in the case of general \(E\) gives sufficient conditions for the sequence of centered and normed linear functionals of the point field to converge in distribution to a Gaussian random value. The second one for \(E= R^d\) \((d\geq 2)\) and “almost” homogeneous kernel \(K\) gives conditions for the weak convergence of the sequence of the rescaled point fields to the Gaussian white noise. And the third one for \(E=R^1\) and homogeneous kernel \(K\) asserts the weak convergence of a sequence of rescaled point processes to a self-similar Gaussian process with the spectral density \(| k|^\alpha\) \((0< \alpha< 1)\).